1. ## Vector Spaces

Which of the following sets of vectors $\alpha=(a_1,a_2,a_3,....,a_n)$ in $R^n$ are subspaces of $R^n$ (n $\geq3)?$ All $\alpha$ such that

(a) $a_1+3a_2=a_3$
(b) $a_1a_2=0$
(c) $a_2=(a_1)^2$
(d) $a_2$ is rational

2. ## Using Axioms

In vector spaces there are 8 (some say 10) axioms that need to be verified to satisfy the definition. Go through each of these and check to see if they are true.

For example in (a)

if $a1 + 3a2 = a3$ where a1,a2,a3 $\epsilon R^n$

it follows then that these vectors are closed under vector addition and scalar multiplication. Since a3 can be expressed as a linear combination of a1 and a2 and a3 $\epsilon R^n$ this is a valid vector space.